Curse of Dimensionality

Full grid solution for multi-asset options pricing with tensor networks ArXiv ID: 2601.00009 “View on arXiv” Authors: Lucas Arenstein, Michael Kastoryano Abstract Pricing multi-asset options via the Black-Scholes PDE is limited by the curse of dimensionality: classical full-grid solvers scale exponentially in the number of underlyings and are effectively restricted to three assets. Practitioners typically rely on Monte Carlo methods for computing complex instrument involving multiple correlated underlyings. We show that quantized tensor trains (QTT) turn the d-asset Black-Scholes PDE into a tractable high-dimensional problem on a personal computer. We construct QTT representations of the operator, payoffs, and boundary conditions with ranks that scale polynomially in d and polylogarithmically in the grid size, and build two solvers: a time-stepping algorithm for European and American options and a space-time algorithm for European options. We compute full-grid prices and Greeks for correlated basket and max-min options in three to five dimensions with high accuracy. The methods introduced can comfortably be pushed to full-grid solutions on 10-15 underlyings, with further algorithmic optimization and more compute power. ...

Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Study ArXiv ID: 2311.07231 “View on arXiv” Authors: Unknown Abstract Option pricing, a fundamental problem in finance, often requires solving non-linear partial differential equations (PDEs). When dealing with multi-asset options, such as rainbow options, these PDEs become high-dimensional, leading to challenges posed by the curse of dimensionality. While deep learning-based PDE solvers have recently emerged as scalable solutions to this high-dimensional problem, their empirical and quantitative accuracy remains not well-understood, hindering their real-world applicability. In this study, we aimed to offer actionable insights into the utility of Deep PDE solvers for practical option pricing implementation. Through comparative experiments, we assessed the empirical performance of these solvers in high-dimensional contexts. Our investigation identified three primary sources of errors in Deep PDE solvers: (i) errors inherent in the specifications of the target option and underlying assets, (ii) errors originating from the asset model simulation methods, and (iii) errors stemming from the neural network training. Through ablation studies, we evaluated the individual impact of each error source. Our results indicate that the Deep BSDE method (DBSDE) is superior in performance and exhibits robustness against variations in option specifications. In contrast, some other methods are overly sensitive to option specifications, such as time to expiration. We also find that the performance of these methods improves inversely proportional to the square root of batch size and the number of time steps. This observation can aid in estimating computational resources for achieving desired accuracies with Deep PDE solvers. ...